A phys­ic­ally nat­ur­al po­ten­tial en­ergy for simple closed curves in \( \mathbb{R}^3 \) is shown to be in­vari­ant un­der Möbi­us trans­form­a­tions. This leads to the rap­id res­ol­u­tion of sev­er­al open prob­lems: round circles are pre­cisely the ab­so­lute min­ima for en­ergy; there is a min­im­um en­ergy threshold be­low which knot­ting can­not oc­cur; min­im­izers with­in prime knot types ex­ist and are reg­u­lar. Fi­nally, the num­ber of knot types with en­ergy less than any con­stant \( M \) is es­tim­ated.

In the to­po­lo­gic­al cat­egory, it is shown that the di­men­sion 4 disk the­or­em holds without fun­da­ment­al group re­stric­tion after sta­bil­iz­ing with many cop­ies of com­plex pro­ject­ive space. As co­rol­lar­ies, a stable 4-di­men­sion­al sur­gery the­or­em and a stable 5-di­men­sion­al \( s \)-cobor­d­ism are ob­tained. These res­ults con­trast with the smooth cat­egory where the use­ful­ness of adding \( \mathbb{C}P^2 \)’s de­pends on chir­al­ity.

These notes de­rive from lec­tures giv­en at UC­SD in 1990. The pur­pose of those lec­tures was to elu­cid­ate the two main the­or­ems of con­trolled lin­ear al­gebra: the van­ish­ing the­or­em for White­head and \( K_0 \)-type ob­struc­tions when only “simply con­nec­ted dir­ec­tions” are left un­con­trolled and the squeez­ing prin­ciple — that once a threshold level of geo­met­ric con­trol is ob­tained any finer amount of con­trol is also avail­able. These notes present in de­tail (and with per­haps some new es­tim­ates on the op­tim­al re­la­tions between \( \varepsilon \), \( \delta \) and di­men­sion) the van­ish­ing the­or­em of Frank Quinn’s Sec­tion 8 [1979] in the con­text of White­head tor­sion.

We show that the to­po­lo­gic­al mod­u­lar func­tor from Wit­ten–Chern–Si­mons the­ory is uni­ver­sal for quantum com­pu­ta­tion in the sense that a quantum cir­cuit com­pu­ta­tion can be ef­fi­ciently ap­prox­im­ated by an in­ter­twin­ing ac­tion of a braid on the func­tor’s state space. A com­pu­ta­tion­al mod­el based on Chern–Si­mons the­ory at a fifth root of unity is defined and shown to be poly­no­mi­ally equi­val­ent to the quantum cir­cuit mod­el. The chief tech­nic­al ad­vance: the dens­ity of the ir­re­du­cible sec­tors of the Jones rep­res­ent­a­tion has to­po­lo­gic­al im­plic­a­tions which will be con­sidered else­where.

Quantum com­puters will work by evolving a high tensor power of a small (e.g. two) di­men­sion­al Hil­bert space by loc­al gates, which can be im­ple­men­ted by ap­ply­ing a loc­al Hamilto­ni­an \( H \) for a time \( t \). In con­trast to this quantum en­gin­eer­ing, the most
ab­stract reaches of the­or­et­ic­al phys­ics has spawned “to­po­lo­gic­al mod­els” hav­ing a fi­nite
di­men­sion­al in­tern­al state space with no nat­ur­al tensor product struc­ture and in which
the evol­u­tion of the state is dis­crete, \( H \equiv 0 \). These are called to­po­lo­gic­al quantum field the­or­ies (TQFTs). These exot­ic phys­ic­al sys­tems are proved to be ef­fi­ciently sim­u­lated on a quantum com­puter. The con­clu­sion is two-fold:

TQFTs can­not be used to define a mod­el of com­pu­ta­tion stronger than the usu­al
quantum mod­el “BQP”.

TQFTs provide a rad­ic­ally dif­fer­ent way of look­ing at quantum com­pu­ta­tion. The
rich math­em­at­ic­al struc­ture of TQFTs might sug­gest a new quantum al­gorithm.

The the­ory of quantum com­pu­ta­tion can be con­struc­ted from the ab­stract study of any­on­ic sys­tems. In math­em­at­ic­al terms, these are unit­ary to­po­lo­gic­al mod­u­lar func­tors. They un­der­lie the Jones poly­no­mi­al and arise in Wit­ten–Chern–Si­mons the­ory. The braid­ing and fu­sion of any­on­ic ex­cit­a­tions in quantum Hall elec­tron li­quids and \( 2D \)-mag­nets are modeled by mod­u­lar func­tors, open­ing a new pos­sib­il­ity for the real­iz­a­tion of quantum com­puters. The chief ad­vant­age of any­on­ic com­pu­ta­tion would be phys­ic­al er­ror cor­rec­tion: An er­ror rate scal­ing like \( e^{-\alpha l} \), where \( l \) is a length scale, and \( \alpha \) is some pos­it­ive con­stant. In con­trast, the “pre­sumptive” qubit-mod­el of quantum com­pu­ta­tion, which re­pairs er­rors com­bin­at­or­ic­ally, re­quires a fant­ast­ic­ally low ini­tial er­ror rate (about \( 10^{-4} \)) be­fore com­pu­ta­tion can be sta­bil­ized.

We de­scribe a class of par­ity- and time-re­versal-in­vari­ant to­po­lo­gic­al states of mat­ter which can arise in cor­rel­ated elec­tron sys­tems in \( 2+1 \)-di­men­sions. These states are char­ac­ter­ized by particle-like ex­cit­a­tions ex­hib­it­ing exot­ic braid­ing stat­ist­ics. \( P \) and \( T \) in­vari­ance are main­tained by a ‘doub­ling’ of the low-en­ergy de­grees of free­dom which oc­curs nat­ur­ally without doub­ling the un­der­ly­ing mi­cro­scop­ic de­grees of free­dom. The simplest ex­amples have been the sub­ject of con­sid­er­able in­terest as pro­posed mech­an­isms for high-\( T_c \) su­per­con­duct­iv­ity. One is the ‘doubled’ ver­sion of the chir­al spin li­quid. The chir­al spin li­quid gives rise to any­on su­per­con­duct­iv­ity at fi­nite dop­ing and the cor­res­pond­ing field the­ory is \( U(1) \) Chern–Si­mons the­ory at coup­ling con­stant \( m=2 \). The ‘doubled’ the­ory is two cop­ies of this the­ory, one with \( m=2 \) the oth­er with \( m=-2 \). The second ex­ample cor­res­ponds to \( Z_2 \) gauge the­ory, which de­scribes a scen­ario for spin-charge sep­ar­a­tion. Our main con­cern, with an eye to­wards ap­plic­a­tions to quantum com­pu­ta­tion, are rich­er mod­els which sup­port non-Abeli­an stat­ist­ics. All of these mod­els, rich­er or poorer, lie in a tightly or­gan­ized dis­crete fam­ily in­dexed by the Baraha num­bers, \( 2\cos(\pi/(k+2)) \), for pos­it­ive in­teger \( k \). The phys­ic­al in­fer­ence is that a ma­ter­i­al mani­fest­ing the \( Z_2 \) gauge the­ory or a doubled chir­al spin li­quid might be eas­ily altered to one cap­able of uni­ver­sal quantum com­pu­ta­tion. These phases of mat­ter have a field-the­or­et­ic de­scrip­tion in terms of gauge the­or­ies which, in their in­frared lim­its, are to­po­lo­gic­al field the­or­ies. We mo­tiv­ate these gauge the­or­ies us­ing a par­ton mod­el or slave-fer­mi­on con­struc­tion and show how they can be solved ex­actly. The struc­ture of the res­ult­ing Hil­bert spaces can be un­der­stood in purely com­bin­at­or­i­al terms. The highly con­strained nature of this com­bin­at­or­i­al con­struc­tion, phrased in the lan­guage of the to­po­logy of curves on sur­faces, lays the ground­work for a strategy for con­struct­ing mi­cro­scop­ic lat­tice mod­els which give rise to these phases.

We dis­cuss gen­er­al­iz­a­tions of quantum spin Hamilto­ni­ans us­ing any­on­ic de­grees of free­dom. The simplest mod­el for in­ter­act­ing any­ons en­er­get­ic­ally fa­vors neigh­bor­ing any­ons to fuse in­to the trivi­al (“iden­tity”) chan­nel, sim­il­ar to the quantum Heis­en­berg mod­el fa­vor­ing neigh­bor­ing spins to form spin sing­lets. Nu­mer­ic­al sim­u­la­tions of a chain of Fibon­acci any­ons show that the mod­el is crit­ic­al with a dy­nam­ic­al crit­ic­al ex­po­nent \( z=1 \), and de­scribed by a two-di­men­sion­al (\( 2D \)) con­form­al field the­ory with cent­ral charge \( c=7/10 \). An ex­act map­ping of the any­on­ic chain onto the \( 2D \) tricrit­ic­al Ising mod­el is giv­en us­ing the re­stric­ted-sol­id-on-sol­id rep­res­ent­a­tion of the Tem­per­ley–Lieb al­gebra. The gap­less­ness of the chain is shown to have to­po­lo­gic­al ori­gin.

Four­i­er trans­form is an es­sen­tial in­gredi­ent in Shor’s factor­ing al­gorithm. In the stand­ard quantum cir­cuit mod­el with the gate set \( \{U(2) \), con­trolled-NOT\( \} \), the dis­crete Four­i­er trans­forms \( F_N=(\omega^{ij})_{N\times N} \) for \( i,j=0,1,\dots,N{-}1 \) and \( \omega=e^{2\pi i/N} \) can be real­ized ex­actly by quantum cir­cuits of size \( O(n^2) \) with \( n=\ln N \), and so can the dis­crete sine or co­sine trans­forms. In to­po­lo­gic­al quantum com­put­ing, the simplest uni­ver­sal to­po­lo­gic­al quantum com­puter is based on the Fibon­acci \( (2+1) \)-to­po­lo­gic­al quantum field the­ory (TQFT), where the stand­ard quantum cir­cuits are re­placed by unit­ary trans­form­a­tions real­ized by braid­ing con­form­al blocks. We re­port here that the large Four­i­er trans­forms \( F_N \) and the dis­crete sine or co­sine trans­forms can nev­er be real­ized ex­actly by braid­ing con­form­al blocks for a fixed TQFT. It fol­lows that an ap­prox­im­a­tion is un­avoid­able in the im­ple­ment­a­tion of Four­i­er trans­forms by braid­ing con­form­al blocks.

Xiao-Song stud­ied un­der Mi­chael Freed­man from 1984–1988 at UC­SD. In Septem­ber 2006, Freed­man wrote Xiao-Song on the in­side cov­er of the So­viet book, How the Steel Was Tempered, to en­cour­age him. Then in Decem­ber 2006, he wrote Xiao-Song again dur­ing the crit­ic­al junc­ture of Xiao-Song’s life. These are the let­ters.

We dis­cuss Hil­bert spaces spanned by the set of string nets, i.e. trivalent graphs, on a lat­tice. We sug­gest some routes by which such a Hil­bert space could be the low-en­ergy sub­space of a mod­el of quantum spins on a lat­tice with short-ranged in­ter­ac­tions. We then ex­plain con­di­tions which a Hamilto­ni­an act­ing on this string net Hil­bert space must sat­is­fy in or­der for the sys­tem to be in the DFib (Doubled Fibon­acci) to­po­lo­gic­al phase, that is, be de­scribed at low en­ergy by an \( \mathit{SO}(3)_3\times\mathit{SO}(3)_3 \) doubled Chern–Si­mons the­ory, with the ap­pro­pri­ate non-abeli­an stat­ist­ics gov­ern­ing the braid­ing of the low-ly­ing qua­si­particle ex­cit­a­tions (nona­beli­ons). Us­ing the string net wave­func­tion, we de­scribe the prop­er­ties of this phase. Our dis­cus­sion is in­formed by map­pings of string net wave­func­tions to the chro­mat­ic poly­no­mi­al and the Potts mod­el.

In a re­cent pa­per, Teo and Kane Phys. Rev. Lett. 104 046401 (2010) pro­posed a three-di­men­sion­al (3D) mod­el in which the de­fects sup­port Ma­jor­ana fer­mi­on zero modes. They ar­gued that ex­chan­ging and twist­ing these de­fects would im­ple­ment a set \( R \) of unit­ary trans­form­a­tions on the zero-mode Hil­bert space which is a “ghostly” re­col­lec­tion of the ac­tion of the braid group on Ising any­ons in two di­men­sions. In this pa­per, we find the group \( T_{2n} \), which gov­erns the stat­ist­ics of these de­fects by ana­lyz­ing the to­po­logy of the space \( K_{2n} \) of con­fig­ur­a­tions of \( 2n \) de­fects in a slowly spa­tially vary­ing gapped free-fer­mi­on Hamilto­ni­an: \( T_{2n}\equiv \pi_1(K_{2n}) \). We find that the group \( T_{2n}=\mathbb{Z}\times T_{2n}^r \), where the “rib­bon per­muta­tion group” \( T_{2n}^r \) is a mild en­hance­ment of the per­muta­tion group
\[ S_{2n}: T_{2n}^r\equiv \mathbb{Z}_2\rtimes E((\mathbb{Z}_2)^{2n}\rtimes S_{2n}) .\]
Here, \( E((\mathbb{Z}_2)^{2n}\rtimes S_{2n}) \) is the “even part” of \( (\mathbb{Z}_2)^{2n}\rtimes S_{2n} \), namely, those ele­ments for which the total par­ity of the ele­ment in \( (\mathbb{Z}_2)^{2n} \) ad­ded to the par­ity of the per­muta­tion is even. Sur­pris­ingly, \( R \) is only a pro­ject­ive rep­res­ent­a­tion of \( T_{2n} \), a pos­sib­il­ity pro­posed by Wil­czek [hep-th/9806228]. Thus, Teo and Kane’s de­fects real­ize pro­ject­ive rib­bon per­muta­tion stat­ist­ics, which we show to be con­sist­ent with loc­al­ity. We ex­tend this phe­nomen­on to oth­er di­men­sions, codi­men­sions, and sym­metry classes. We note that our ana­lys­is ap­plies to 3D net­works of quantum wires sup­port­ing Ma­jor­ana fer­mi­ons; thus, these net­works are not re­quired to be planar. Be­cause it is an es­sen­tial in­put for our cal­cu­la­tion, we re­view the to­po­lo­gic­al clas­si­fic­a­tion of gapped free-fer­mi­on sys­tems and its re­la­tion to Bott peri­od­icity.

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